Types of matrices
Definition.A square matrix is a matrix in which the number of rows equals the number of columns (size n×n), the number n, is called the order of the matrix.
|4||1||-7||- a square matrix with size 3×3|
Definition.The zero matrix is a matrix whose entries are equal to zero, ie, aij = 0, ∀i, j.
|0||0||0||- zero matrix|
Definition.Row vector is a matrix consisting of a one row.
|1||4||-5||- row vector|
Definition.Column vector is a matrix consisting of a one column.
|8||- column vector|
Definition.Diagonal matrix is a square matrix whose entries standing outside the main diagonal are equal zero.
Example of diagonal matrix.
|4||0||0||- diagonal entries are arbitrarynot diagonal entries are equal to zero|
Definition.Identity matrix is a diagonal matrix in which all the elements on the main diagonal are equal to 1.
Denote.The identity matrix usually denoted by I.
Example of identity matrix.
|I =||1||0||0||- diagonal entries are equal to 1not diagonal entries are equal to 0|
Definition.Upper triangular matrix is a matrix whose elements below the main diagonal are equal to zero.
Example of upper triangular matrix.
Definition.Lower triangular matrix is a matrix whose elements above the main diagonal are equal to zero.
Example of lower triangular matrix.
N.B. The diagonal matrix is a matrix that is both upper triangular and lower triangular.
Definition.Specifically, a matrix is in row echelon form if:
- all nonzero rows are above all zero rows;
- if the first non-zero element of a row is located in a column with the number i, and the next line is not zero, then the first non-zero element of the next line should be in the column with number greater than i.
Examples of row echelon form of matrix.
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